HHigher order antibunching in intermediate statesAmit Verma , Navneet K Sharma and Anirban Pathak Department of Physics, JIIT University, A-10, Sectror-62, Noida, UP-201307, India.
Abstract
Since the introduction of binomial state as an intermediate state, di(cid:27)erent intermediate stateshave been proposed. Di(cid:27)erent nonclassical e(cid:27)ects have also been reported in these intermediatestates. But till now higher order antibunching or higher order subpoissonian photon statisticsis predicted only in one type of intermediate state, namely shadowed negative binomial state.Recently we have shown the existence of higher order antibunching in some simple nonlinearoptical processes to establish that higher order antibunching is not a rare phenomenon (
J. Phys.B 39 (2006) 1137 ). To establish our earlier claim further, here we have shown that the higherorder antibunching can be seen in di(cid:27)erent intermediate states, such as binomial state, reciprocalbinomial state, hypergeometric state, generalized binomial state, negative binomial state andphoton added coherent state. We have studied the possibility of observing the higher ordersubpoissonian photon statistics in di(cid:27)erent limits of intermediate states. The e(cid:27)ect of di(cid:27)erentcontrol parameters have also been studied in this connection and it has been shown that thedepth of nonclassicality can be tuned by controlling various physical parameters. IntroductionAn intermediate state is a quantum state which reduces to two or more distinguishably di(cid:27)erent states(normally, distinguishable in terms of photon number distribution) in di(cid:27)erent limits. In 1985, such a statewas (cid:28)rst time introduced by Stoler et al. [1]. To be precise, they introduced Binomial state (BS) as a statewhich is intermediate between the most nonclassical number state | n (cid:105) and the most classical coherent state | α (cid:105) . They de(cid:28)ned BS as | p, M (cid:105) = (cid:80) Mn =0 B Mn | n (cid:105) = M (cid:88) n =0 (cid:113) M C n p n (1 − p ) M − n | n (cid:105) ≤ p ≤ . (1)This state is called intermediate state as it reduces to number state in the limit p → and p → (as | , M (cid:105) = | (cid:105) and | , M (cid:105) = | M (cid:105) ) and in the limit of M → ∞ , p → , where α is a real constant, it reduces toa coherent state with real amplitude. Since the introduction of BS as an intermediate state, it was alwaysbeen of interest to quantum optics, nonlinear optics, atomic physics and molecular physics community.Consequently, di(cid:27)erent properties of binomial states have been studied [2-7]. In these studies it has beenobserved that the nonclassical phenomena (such as, antibunching, squeezing and higher order squeezing)can be seen in BS. This trend of search for nonclassicality in Binomial state, continued in nineties. In onehand, several versions of generalized BS have been proposed [3-5] and in the other hand, people went beyondbinomial states and proposed several other form of intermediate states (such as, excited binomial state[6], odd excited binomial state [7], hypergeometric state [8], negative hypergeometric state [9], reciprocalbinomial state [10], shadowed state [11], shadowed negative binomial state [12] and photon added coherentstate [13] etc.). The studies in the nineties were mainly limited to theoretical predictions but the recentdevelopments in the experimental techniques made it possible to verify some of those theoretical predictions.For example, we can note that, as early as in 1991 Agarwal and Tara [13] had introduced photon addedcoherent state as | α, m (cid:105) = a † m | α (cid:105)(cid:104) a m a † m (cid:105) , (2) [email protected] [email protected] [email protected] The state is named as binomial state because the photon number distribution associated with this state ` i.e. | B Mn | ´ issimply a binomial distribution. a r X i v : . [ qu a n t - ph ] J un where m is an integer and | α (cid:105) is coherent state) but the experimental generation of the state has happenedonly in recent past when Zavatta, Viciani and Bellini [14] succeed to produce it in 2004. It is easy toobserve that (2) represents an intermediate state, since it reduces to coherent state in the limit m → andto number state in the limit α → . This state can be viewed as a coherent state in which additional m photon are added. The photon number distribution of all the above mentioned states are di(cid:27)erent but allthese states belong to a common family of states called intermediate state. It has also been found that mostof these intermediate states show antibunching, squeezing, higher order squeezing, subpoissonian photonstatistics etc. but higher order antibunching has been reported only in shadowed negative binomial state[12]. Inspired by these observations, many schemes to generate intermediate states have been proposed inrecent past [14-17].The reason behind the study of nonclassical properties of intermediate states lies in the fact that the mostof the interesting recent developments in quantum optics have arisen through the nonclassical properties ofthe radiation (cid:28)eld only. For example, antibunching and squeezing, which do not have any classical analogue[18-20], have extensively been studied in last thirty years. But the majority of these studies are focusedon lowest order nonclassical e(cid:27)ects. Higher order extensions of these nonclassical states have only beenintroduced in recent past [21-24]. Among these higher order nonclassical e(cid:27)ects, higher order squeezing hasalready been studied in detail [21, 22, 25, 26] but the higher order antibunching (HOA) is not yet studiedrigourously.The idea of HOA was introduced by Lee in a pioneering paper [23] in 1990, since then it has beenpredicted in two photon coherent state [23], shadowed negative binomial state [12], trio coherent state [27]and in the interaction of intense laser beam with an inversion symmetric third order nonlinear medium [28].From the fact that in (cid:28)rst 15 years after its introduction, HOA was reported only in some particular cases,HOA appeared to be a very rare phenomenon. But recently we have shown that the HOA is not a rarephenomenon [29] and it can be seen in simple optical processes like six wave mixing process, four wavemixing process and second harmonic generation. To establish that further, here we have shown the existenceof HOA in di(cid:27)erent intermediate states, namely, binomial state, reciprocal binomial state, photon addedcoherent state, hypergeometric state, Roy-Roy generalized binomial state and negative binomial state.The present work is motivated by the recent experimental observation of intermediate state [14], theo-retical observation of possibility of observing HOA in some simple optical systems [29] and the fact that theintermediate states, which frequently show di(cid:27)erent kind of nonclassicality, form a big family of quantumstate. But till now HOA has been predicted only in one member (Shadowed negative binomial state) of sucha big family of quantum states [12]. Motivated by these facts the present work aims to study the possibilityof HOA in all the popularly known intermediate states. The theoretical predictions of the present studycan be experimentally veri(cid:28)ed with the help of various intermediate state generation schemes and homodyneexperiment, since the criteria for HOA appears in terms of factorial moment, which can be measured by usinghomodyne photon counting experiments [30-33]. In the next section we have brie(cid:29)y described the criteriaof HOA. In section 3 we have shown that the HOA of any arbitrary order can be seen in BS. In section 4,Roy-Roy generalized binomial state [4] is studied and existence of HOA is predicted. Calculational detailsand methodology have been shown only in algebraically simple cases which are described in section 3 and 4.Section 5 is divided in several subsections and we have followed the same procedure and have studied thepossibilities of observing HOA in di(cid:27)erent intermediate states namely, reciprocal binomial state, negativebinomial state, hypergeometric states and photon added coherent state. One subsection is dedicated for thediscussion of one particular intermediate state. Finally section 6 is dedicated to conclusions. Criteria of HOAThe criterion of HOA is expressed in terms of higher order factorial moments of number operator. There existseveral criterion for the same which are essentially equivalent. Initially, using the negativity of P function[18], Lee introduced the criterion for HOA as R ( l, m ) = (cid:68) N ( l +1) x (cid:69) (cid:68) N ( m − x (cid:69)(cid:68) N ( l ) x (cid:69) (cid:68) N ( m ) x (cid:69) − < , (3)2here N is the usual number operator, (cid:10) N ( i ) (cid:11) = (cid:104) N ( N − ... ( N − i + 1) (cid:105) is the ith factorial moment ofnumber operator, (cid:104)(cid:105) denotes the quantum average, l and m are integers satisfying the conditions ≤ m ≤ l and the subscript x denotes a particular mode. Ba An [27] choose m = 1 and reduced the criterion of l thorder antibunching to A x,l = (cid:68) N ( l +1) x (cid:69)(cid:68) N ( l ) x (cid:69) (cid:104) N x (cid:105) − < (4)or, (cid:68) N ( l +1) x (cid:69) < (cid:68) N ( l ) x (cid:69) (cid:104) N x (cid:105) . (5)Physically, a state which is antibunched in l th order has to be antibunched in ( l − th order. Therefore, wecan further simplify (5) as (cid:68) N ( l +1) x (cid:69) < (cid:68) N ( l ) x (cid:69) (cid:104) N x (cid:105) < (cid:68) N ( l − x (cid:69) (cid:104) N x (cid:105) < (cid:68) N ( l − x (cid:69) (cid:104) N x (cid:105) < ... < (cid:104) N x (cid:105) l +1 (6)and obtain the condition for l − th order antibunching as d ( l ) = (cid:68) N ( l +1) x (cid:69) − (cid:104) N x (cid:105) l +1 < . (7)This simpli(cid:28)ed criterion (7) coincides exactly with the physical criterion of HOA introduced by Pathakand Garica [28] and the criterion of Erenso, Vyas and Singh [34], recently Vogel has reported a class ofnonclassicality conditions based on higher order factorial moments [35]. All these criteria essentially lead tosame kind of nonclassicality which belong to the class of strong nonclassicality according to the classi(cid:28)cationscheme of Arvind et al [36]. Here we can note that d ( l ) = 0 and d ( l ) > corresponds to higher ordercoherence and higher order bunching (many photon bunching) respectively. Actually, (cid:10) a † l a l (cid:11) = (cid:10) N ( l ) (cid:11) isa measure of the probability of observing l photons of the same mode at a particular point in space timecoordinate. Therefore the physical meaning of inequalities (6) is that the probability of detection of singlephoton pulse is greater than that of a two photon in a bunch and that is greater than the probability ofdetection of three photon in a bunch and so on. This is exactly the characteristic that is required in aprobabilistic single photon source used in quantum cryptography. In other words all the probabilistic singlephoton sources used in quantum cryptography should satisfy the criteria (7) of HOA [37]. Binomial state is originally de(cid:28)ned as (1), from which it is straight forward to see that a | p, M (cid:105) = (cid:80) Mn =0 (cid:110) M !( n − M − n )! p n (1 − p ) M − n (cid:111) | n − > = (cid:80) M − l =0 (cid:110) M ( M − l !( M − − l )! p l +1 (1 − p ) M − − l (cid:111) | l > ( assuming n − l )= [ M p ] (cid:80) M − l =0 (cid:110) ( M − l !( M − − l )! p l (1 − p ) M − − l (cid:111) | l > = [ M p ] | p, M − (cid:105) . (8)Similarly, we can write, a | p, M (cid:105) = [ M ( M − p ] | p, M − (cid:105) a | p, M (cid:105) = [ M ( M − M − p ] | p, M − (cid:105) ... ... ... a l | p, M (cid:105) = [ M ( M − .... ( M − l + 1) p l ] | p, M − l (cid:105) = (cid:104) M !( M − l )! p l (cid:105) | M − l, p (cid:105) . (9)Therefore, (cid:104) M, p | a † l = (cid:104) M − l, p | (cid:20) M !( M − l )! p l (cid:21) (10)3nd consequently, (cid:104) M, p | n ( l ) | p, M (cid:105) = (cid:104) M, p | a † l a l | p, M (cid:105) = (cid:20) M !( M − l )! p l (cid:21) . (11)Now substituting (11) in equation (7) we obtain the condition for lth order antibunching as d ( l ) = (cid:20) M !( M − l − p l +1 (cid:21) − [ M p ] l +1 < (12)or, ( M − M − .... ( M − l ) < M l (13)which is always satis(cid:28)ed for any M > l and both M and l are positive (since every term in left is < M ). As M is the number of photons present in the (cid:28)eld and d ( l ) is a measure of correlation among ( l + 1) photons,therefore M ≥ ( l + 1) or M > l . Consequently, a binomial state always shows HOA and the highest possibleorder of antibunching that can be seen in a binomial state is equal to M − , where M is the number ofphoton present in the (cid:28)eld. From (12) it is straight forward to see that the number state is always higherorder antibunched and in the other extreme limit (when p → , M → ∞ and the BS reduces to coherentstate) d ( l ) = 0 , which is consistent with the physical expectation. We have already mentioned that there are di(cid:27)erent form of generalized binomial states [3-5]. For the presentstudy we have chosen generalized binomial state introduced by Roy and Roy [4]. They have introduced thegeneralized binomial state (GBS) as | N, α, β (cid:105) = N (cid:88) n = o (cid:112) ω ( n, N, α, β ) | n (cid:105) (14)where, ω ( n, N, α, β ) = N !( α + β + 2) N ( α + 1) n ( β + 1) N − n n !( N − n )! (15)with α, β > − , n = 0 , , ...., N , and ( a ) = 1 ( a ) n = a ( a + 1) .... ( a + n − . (16)This intermediate state reduces to vacuum state, number state, coherent state, binomial state and negativebinomial state in di(cid:27)erent limits of α , β and N . In order to obtain an analytic expression of d ( l ) for thisparticular generalized binomial state we need to prove following useful identity:Identity1: a ( a + 1) n = ( a ) n +1 (17)Proof: Using (16) we can write ( a + 1) n = ( a + 1) ... ( a + n ) = a ( a + 1) ... ( a + n ) a = ( a ) n +1 a . Therefore, a ( a + 1) n = ( a ) n +1 . Now it is easy to see that the above identity (17) yields the following useful relations: ( α + 1) l +1 = ( α + 1)( α + 2) l (18)and ( α + β + 2) N = ( α + β + 2)( α + β + 3) N − = ( α + β + 2)( α + 2 + β + 1) N − . (19)4sing (14) and (15) we can obtain a | N, α, β (cid:105) = (cid:80) Nn =0 (cid:110) N !( α + β +2) N ( α +1) n ( n − β +1) N − n ( N − n )! (cid:111) | n − (cid:105) = (cid:80) N − l =0 (cid:110) N ( N − α + β +2) N ( α +1) l +1 l ! ( β +1) N − − l ( N − − l )! (cid:111) | l (cid:105) , (20)where n = l − has been used. Now we can apply (18) and (19) on (20) to obtain a | N, α, β (cid:105) = (cid:110) N ( α +1)( α + β +2) (cid:111) (cid:80) N − l =0 (cid:110) ( N − α +2) l ( β +1) N − − l ( α +2+ β +1) N − l !( N − − l )! (cid:111) | l (cid:105) = (cid:110) N ( α +1)( α + β +2) (cid:111) (cid:80) N − n =0 (cid:112) ω ( n, N − , α + 1 , β ) | n (cid:105) , (21)where dummy variable l is replaced by n. Therefore, (cid:104)